Projectile guidance with accelerometers and a GPS receiver

ABSTRACT

A projectile guidance system without gyros in which the projectile has an orthogonal body coordinate system. The projectile has a triax of accelerometers providing x, y and z acceleration data measured along the x, y and z axes respectively. A GPS antenna and receiver means provides onboard GPS position and velocity data in earth referenced navigational coordinates. A computer and program means stores and accesses time indexed GPS position and GPS velocity data and transforms x, y and z axis acceleration data from body to navigation coordinates. The program means is responsive to corresponding time indexed acceleration data and to GPS velocity and position data for calculating and outputting an estimated projectile roll, pitch and yaw angle via optimal smoothing techniques with respect to local level for each time index iteration of present position to a flight control system, which actuates a divert propulsion system for guiding the projectile to a predetermined location.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to the field of aircraft or projectile guidanceusing divert propulsion means; the propulsion means using aerodynamicmaneuvering or gas propulsion to adjust the flight path of a projectilein response to a command from a guidance system; the guidance systemincluding an attitude determination and estimation means such as a GPSsignal and configured to respond to signals from a first and secondlinear accelerometer for an estimation of the projectile's roll angle,and information from an additional third accelerometer for a pitch angleestimation.

2. Description of the Prior Art

The general problem to be solved is how to guide a gun-fired projectileonto a target with a known geographic location at lowest cost withreasonable reliability. This application addresses a portion of thatproblem by providing various divert propulsion means based onaerodynamic maneuvering and/or gas propulsion to adjust the flight pathof a projectile in response to a command from a guidance system. Thisapplication also provides means for enabling the guidance system todetermine appropriate commands to communicate to the propulsion systembased on an estimate of the trajectory of the projectile from launch,the initial conditions of the projectile immediately after launch andthe distance, direction and altitude to the idealized trajectorycalculated for the projectile at the time of launch or firing.

Prior approaches to the problem to be solved used external aerodynamicsurfaces, e.g., canards, controlled by a guidance system that reliedupon only a GPS receiver and a turns counter. Unfortunately, however,external surfaces cause difficulties in launching the projectiles, e.g.,from within a launch barrel, and the GPS signals may be vulnerable tojamming. Accordingly, where external aerodynamic surfaces, e.g.,canards, are used, additional provisions must be made to minimize launchtube (barrel) interference and to accommodate large forces imposed onthe canards due to projectile acceleration and aerodynamic drag.Furthermore, additional measures may need to be implemented to ensurereliable reception and processing of an uncorrupted GPS signal, e.g.,protection of the GPS signal.

Additional approaches for controlling the aerodynamic surfaces haverelied upon a GPS receiver, a turns counter, and a triax of gyros, whichfurther increases the cost of the system. Yet, this approach remainsvulnerable to corruption of the GPS signal because, as in the firstapproach described above, in the event the GPS signal is lost, nomechanism exists to account for any external forces that may act on theprojectile throughout its flight.

Prior attempts to address the vulnerability to GPS signal jamming havefocused on either preventing interference with the GPS signal orenabling operation with limited GPS data. Attempts to enable operationwith limited GPS data have typically involved increasing the performanceand/or functionality of the inertial instruments on board theprojectile. For example, to address the inability of the above-mentionedsecond approach to account for the projectile's reaction to externalforces if/when the GPS signal may be lost, additional accelerometers maybe incorporated into the system to enable compensation for theprojectile's reaction to external forces, i.e., the sensor package maycomprise a complete IMU. Gyroscopic instruments for aircraft use arewell known and available in a number of technologies such as iron rotorand tuned rotor gyros, ring laser gyros, multi-oscillator gyros, zerolock gyros (ZLG), fiber optic gyros, resonator gyros such as HRGs orhemispherical or tubular ceramic resonant gyros and the like.

Unfortunately, however, incorporation of additional gyroscopicinstruments and/or accelerometers are considered much too costly becauseof the gyros already in the package. Moreover, use of such instrumentsimposes additional constraints on the operational envelopes of theprojectiles. For example, gyroscopic instruments are typically subjectto failure modes and uncertainties relating to launch accelerations inthe range of 15,000-30,000 Gs. Further, use of these technologiesusually requires that the vehicle carry at least one gyro in a gimbaledor strap-down arrangement with the attendant disadvantages of cost,weight and power dissipation.

Accordingly, a need exists for a system and process for guiding aprojectile to a target while eliminating reliance on aerodynamicsurfaces external to the projectile and while minimizing or eliminatingthe vulnerability of the system and process to interference with, e.g.,jamming of, the GPS signal. Thus, it would be advantageous to have animproved, cost-effective system and process for providing projectileguidance in the presence of GPS signal jamming and/or with limited GPSdata.

SUMMARY OF THE INVENTION

In accordance with a first exemplary embodiment of the invention systemand process, a projectile without gyros is guided toward a target usingdivert propulsion means substantially internal to the projectile. Thedivert propulsion means are controlled by a guidance system relying on aGPS signal and a triax of accelerometers. The guidance system uses acomputer-implemented process that responds to GPS position and GPS deltavelocity data along with data from the accelerometer triax to determinethe projectile's estimated attitude in pitch, roll and yaw. With theprojectile's position known from data provided by the GPS signalreceived by a GPS receiver, the computer-implemented process determinesthe projectile's attitude in navigational coordinates and creates a timeindexed record of the projectile's trajectory after the on-board GPSreceiver locks on to the required number of satellites. The data in thetime indexed record of the trajectory is filtered and smoothed. As theprojectile rolls, the accelerometers are used to measure the forcesacting on the projectile and the projectile's rotation rates. In somealternative embodiments of the invention, the GPS is used to providedata for the calculation of initial aiming and velocity errors and forthe calibration of the accelerometers when positioned in the gun barrelprior to launch.

In an exemplary embodiment, the divert propulsion means comprises one ormore aerodynamic surfaces located substantially within the projectile.In accordance with this embodiment, an exemplary projectile has one ormore supply ports located so as to receive a supply of gas having anelevated total pressure (dynamic head), e.g., located on the leadingsurface of the projectile. The projectile further comprises one or moreexit ports situated so as to enable the gas to exit the projectile at alocation where the local static pressure is lower than the pressurerealized at the one or more supply ports, e.g., at the side or trailingedge of the projectile. The supply ports are in fluid communication withthe exit ports through gas metering means and gas directing means whichare both controlled by the guidance system. By manipulating the quantityof gas received through the supply ports and/or by manipulating thedirection and/or velocity of the gas exiting the projectile through theexit ports, the guidance system may effectively guide the projectileonto a target.

In another exemplary embodiment, a projectile similarly comprises one ormore exit ports in fluid communication with a gas chamber through asimilar gas metering means and gas directing means. In accordance withthis embodiment, high pressure gas may be supplied via a combustion, orother chemical, process conducted within the projectile, by pre-chargingthe projectile before or during launch, or by any other means known inthe art for producing a compressed volume of fluid. Exemplary methods ofsupplying the gas chamber include chemical reaction, e.g., combustion,of monopropellants, bi-propellants, solid propellants, liquidpropellants, and solid/liquid propellant hybrids. An exemplary methodfor pre-charging the projectile before launch involves loading thechamber with compressed fluid and closing the supply port as soon as thechamber is loaded. An exemplary method of pre-charging the projectileduring launch involves opening a port in the projectile so as to permithigh pressure gas from within the launch tube (e.g., gun barrel) to flowinto the chamber and closing the port before, or very shortly after, theprojectile exits the launch tube, thereby loading the chamber withcompressed gas. Accordingly, the guidance system may manipulate thedirection and/or velocity of the gas exiting the projectile through theexit ports to effectively guide the projectile onto a target.

It should be noted that the cost associated with the guidance system maydepend upon the accuracy of the accelerometers that must be used, whichdepends upon the required delivery accuracy of the projectile. Wherecost must be reduced, lower accuracy accelerometers can be used with agreater reliance on GPS signal data after launch. Therefore, to reducecost while retaining reasonable levels of projectile delivery accuracy,greater reliance must be placed on GPS signal data. Thus, an aspect ofthe present invention provides for improved reliability of GPS signals.

In an exemplary embodiment, GPS signal data may be acquired from GPSsignals transmitted from one or more satellites and received by one ormore antennas on the projectile. Where the potential exists to encounterjamming of one or more of the GPS signals, one or more corresponding GPSsignal jamming detector(s) may be included to monitor each GPS signaland to detect whether each GPS signal has been subjected to jamming. Ifsuch jamming is detected, an anti-jammer will implement GPS signalprotection measures.

Exemplary GPS signal protection measures may include causing theprojectile to engage, or remain engaged, in a periodic motion such asrolling, e.g., via the projectile's divert propulsion means, and usingthe periodic motion of the projectile to selectively sample GPS signalsin such a manner as to detect and omit jammed GPS signals and to enablereliable processing of unjammed signals, e.g., by selectively andperiodically avoiding GPS signals that exhibit jamming. It should benoted that as a projectile proceeds through each cycle of its periodicmotion, i.e., each rolling revolution, each of the one or more antennaswill periodically be positioned to receive a GPS signal from each of theone or more satellites. Thus, so long as at least one GPS signal from asatellite remains capable of being received by an antenna in an unjammedcondition, each of the one or more antennas will periodically be freefrom jamming, e.g., during at least a portion of the roll attitude.

Alternatively, where the projectile motion permits one or more antennasto remain oriented so as to continuously receive an unjammed GPS signal,such as where the projectile is not spinning, exemplary GPS signalprotection measures may include detecting whether a GPS signal receivedby one or more of the antennas is free from jamming and switching a GPSreceiver to use only the unjammed signals. It should be noted that insuch an exemplary embodiment, the projectile need not be spinning orrolling. Accordingly, this protection measure may be implemented withprojectiles that are not roll-stabilized. It should also be noted that,in accordance with this embodiment, the antenna(s) may be decoupled fromthe main body of the projectile via a mechanism such as a slip ring.Accordingly, the projectile may be spinning or roll stabilized while theantennas are maintained in a fixed orientation with respect to one ormore satellites. Similarly, the projectile may be stationary while theantennas undergo periodic motion, e.g., spinning, with respect to one ormore satellites.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective and exploded view of a projectile or unmannedairborne device equipped with an on-board GPS receiver, a set ofaccelerometers coupled to a vehicle frame, a plurality of antennas, andon-board computer cards for executing the invention process;

FIG. 2 shows a launch vehicle and a desired trajectory with a flighterror envelope region around the trajectory;

FIGS. 3 a, 3 b, 3 c and 3 d each show a block diagram for processingaccelerometer and GPS data for the purpose of estimating the trajectoryof flight;

FIG. 4 is a perspective schematic view of an aircraft showing a typicalcoordinate system and defining angles of rotation;

FIG. 5 is a block diagram for the functional steps in a Roll AngleComputation;

FIG. 6 is a flow chart for the steps in a typical Kalman filter and forcalculating an alternate roll angle;

FIG. 7 a is a flow chart of the forward filter steps in a typicalsmoother;

FIG. 7 b is a flow chart of the backward filter steps in a typicalsmoother;

FIG. 8 illustrates time dependent GPS signals as received by a pluralityof antennas on a rotating projectile in the presence of GPS signaljamming;

FIG. 9 illustrates a projectile with external aerodynamic surfaces forguiding the projectile in flight;

FIGS. 10 a, 10 b, and 10 c illustrate side, top, and top views,respectively, of a projectile having one or more supply ports and one ormore exit ports in accordance with an exemplary embodiment of theinvention;

FIGS. 11 a and 11 b illustrate, during and after launch from a gunbarrel, a projectile having a supply port positioned to permit launchgasses to pre-charge the projectile for use in guiding the projectileduring flight in accordance with an exemplary embodiment of theinvention; and

FIGS. 12 a, 12 b, 12 c, 12 d, and 12 e illustrate projectiles inaccordance with exemplary embodiments of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 schematically shows the major components in an exemplaryembodiment of a projectile guidance system without gyros 10 distributedin the exploded perspective view of a projectile 12. The projectile 12is depicted in flight after having been fired from a gun (not shown).

The projectile is shown having an orthogonal body fixed coordinatesystem. The rotation rate components in roll φ, with a longitudinal axisX_(b) a pitch axis Y_(b) and a yaw axis Z_(b). FIG. 4 shows an aircraftwith a corresponding orthogonal body coordinate system. The rotationrate components in roll φ, pitch θ and yaw ψ are defined in FIG. 4. Alocally level fixed earth navigational coordinate system is alsodepicted in FIG. 4. When the wander angle is zero, the Y_(N) axis istypically parallel to a meridian of longitude and points true north. TheX_(N) axis is parallel to a parallel of latitude and points east and theZ_(N) axis extends from the surface of the earth as a gravity vector.The angle α is used in a navigational system to traverse the poles ofthe earth without encountering position singularities. In operation,components of time indexed data elements such as projectile accelerationand velocity are transformed from the body coordinates, indicated by asubscript “b” into the earth referenced coordinate data elementsindicated by a subscript N using conventional direction cosine matrices.

Referring again to FIG. 1, the projectile, guided by the projectileguidance system without gyros, has a triax of accelerometers 14. Thetriax has an x-axis accelerometer for providing x-axis acceleration datameasured along the X_(b) axis, a y-axis-accelerometer for providingy-axis acceleration data measured along the Y_(b) axis and a z-axisaccelerometer for providing z-axis acceleration data measured along theZ_(b) axis. A GPS antenna represented by band 16 and a GPS receiverrepresented by module 18, in combination, represent a means forproviding onboard GPS position and altitude and GPS velocity data inearth referenced navigational coordinates. Alternative antenna patterns88, 89 are configured to receive GPS data from one or more satellites78, 79. While only antenna patterns 88, 89 and antenna band 16 areshown, it should be noted that any number, configuration, and/ororientation of antennas may be included on the projectile, each antennabeing configured and oriented to receive GPS signals from a differentdirection or range of directions (e.g., each antenna pattern having alobe directed toward a different direction or range of directions).Similarly, although only one GPS receiver 18 is shown, it should benoted that a plurality of GPS receivers may be implemented toaccommodate processing of the GPS signals received through the specificnumber, configuration and orientation of antennas implemented on theprojectile.

Module 20 represents a guidance processor which provides the function ofa computer, memory and program means for storing and accessing timeindexed GPS position and GPS velocity data, for implementing GPS signalprotection measures as described herein, and for transforming the x, yand z axis acceleration data from body to navigation coordinates. Theacceleration data is arrayed and has time indexes that are common withthe time indexes for the GPS position and velocity data. A turns counter22 provides an ac signal in flight representing the angular rate ofrotation of the projectile in the earth's magnetic field.

The computer, memory and program means on module 20 receives andresponds to corresponding time indexed acceleration data, GPS velocityand position data for calculating and outputting corresponding sets oftime indexed estimates of the projectile roll, pitch and yaw angles withrespect to the locally level earth referenced navigation coordinates foreach time index iteration of the program to provide present position,velocity and acceleration data to a flight control and ballistic datasystem represented by module 26 for it to use in guiding the projectileto a predetermined target location.

The flight control and ballistic data system controls the projectile byproviding commands to a divert propulsion means, such as externalcontrol surfaces, e.g., canards, 28, 30, to keep the projectile on atrajectory to the target location. Prior approaches to the problem to besolved used external aerodynamic surfaces, e.g., canards, controlled bya guidance system that relied upon only a GPS receiver and a turnscounter. As mentioned above, it has been noted that externallyimplemented control surfaces cause difficulties in launchingprojectiles, e.g., from within a launch barrel. Accordingly, whereexternal aerodynamic surfaces, e.g., canards, are used, additionalprovisions must be made to minimize launch tube (barrel) interferenceand to accommodate large forces imposed on the canards due to projectileacceleration and aerodynamic drag.

FIG. 9 illustrates an exemplary embodiment of a projectile with externalaerodynamic surfaces for guiding the projectile in flight. As shown inFIG. 9, aerodynamic surfaces 28, 30 may be positioned in a variety oflocations about the exterior surfaces of the projectile and may beactuated so that gas flowing past the exterior of the projectile willact on the control surfaces so as change the attitude, i.e.,orientation, of the projectile or to modify the flight path of theprojectile and to guide the projectile to a target.

In addition, this invention provides various internally implementeddivert propulsion means for guiding a projectile to a target. Forexample, as shown in FIGS. 10 a, 10 b, and 10 c, an exemplary projectile1000 may be configured to respond to commands from the guidance systemby admitting a flow of fluid 1001 through the leading portion of theprojectile in substantially the free stream direction 1002, allowing thefluid 1001 to pass through the interior of the projectile 1000, andcausing the fluid 1001 to exit the projectile 1000 in a modifieddirection 1003. In an exemplary embodiment, a projectile comprises oneor more supply ports 1011 located so as to receive a supply of fluid1001 having an elevated total pressure (combined static pressure anddynamic head), e.g., located on the leading surface 1014 of theprojectile 1000. The projectile 1000 further comprises one or more exitports 1031 situated so as to enable the fluid 1001 to exit theprojectile 1000 at a location where the local static pressure is lowerthan the pressure realized at the one or more supply ports 1011, e.g.,at the side 1032 or trailing edge 1033 of the projectile 1000. Thesupply ports 1011 are in fluid communication with the exit ports 1031through gas metering means 1021 and gas directing means 1022 which areboth controlled by the guidance system. It should be noted that gasmetering means 1021 may comprise any valve known in the art forcontrolling the rate of flow (e.g., by varying the cross-sectional areaof a passage) of the fluid (e.g., air, water) through which theprojectile may fly. It should also be noted that gas directing means1022 may comprise an internal control surface, e.g., an internal canardor a flow tube, that, when actuated, may be oriented at an angle to thefree stream direction 1002 so as to direct the flow exiting theprojectile 1000 in a direction other than the free stream direction.

As the momentum of the fluid 1001 flowing within the projectile 1000 ischanged from its initial state where it enters the projectile 1000,i.e., at supply port 1011, to its final state where it exits theprojectile 1000, i.e., at exit port 1031, a force will be exerted on theprojectile 1000, causing the linear momentum of projectile 1000 tochange. If the exit port is positioned, and the direction of exit flowcontrolled, in such a manner that the exit flow vector does not passthrough both the exit port 1031 and the center of gravity 1080 of theprojectile, then a moment will be exerted on the projectile 1000 causingthe angular momentum of projectile 1000 to change, i.e., causing therotation of the projectile to change. It should also be noted that, inaccordance with this embodiment, actuation of gas metering means 1021and gas directing means 1022 may be timed to coincide with one or moredesired orientation of projectile 1000 to achieve a desired change, orcombination of changes, in the flight path and/or orientation ofprojectile 1000. Accordingly, by manipulating the quantity of fluid 1001received through the supply ports 1011, by manipulating the directionand/or velocity of the fluid 1001 exiting the projectile 1000 throughthe exit ports 1031, by positioning the exit ports to align, or notalign, with the center of gravity of the projectile, and/or by timingthe actuation of the divert propulsion means, the guidance system mayeffectively guide the projectile 1000 onto a target.

In another exemplary embodiment, as shown in FIGS. 11 a and 11 b, aprojectile similarly comprises one or more exit ports 1131 in fluidcommunication with a gas chamber through a similar gas metering means1121 and gas directing means 1122. More particularly, FIGS. 11 a and 11b illustrate, during and after launch from a gun barrel, an exemplaryprojectile 1100 having a supply port 1111 positioned to permit launchgasses, e.g., products of combusted gun powder or other propellant, topre-charge the projectile 1100 for use in guiding the projectile 1100during flight.

As shown in FIG. 11 a, a supply port 1111 may be opened while theprojectile 1100 is in the barrel 1177. Accordingly, high pressure launchgasses are permitted to enter and pressurize chamber 1118. Then, asshown in FIG. 11 a, supply port 1111 is closed either before, or veryshortly after, the projectile exits the launch tube, thereby retainingin chamber 1118, a supply of compressed gas. Similar to other exemplaryembodiments described herein, projectile 1100 may comprise one or moreexit ports 1131 in fluid communication with gas chamber 1118 through asimilar gas metering means 1121 and gas directing means 1122.Accordingly, the guidance system may actuate gas metering means 1121 andgas directing means 1122 to manipulate the direction and/or velocity ofthe gas exiting the projectile through the exit ports 1131 toeffectively guide the projectile onto a target. It should be noted thatdifficulties may be encountered where particulates or other undesirableconstituents are permitted to enter chamber 1118 during firing. Toalleviate difficulties, e.g., clogging, fouling, associated with suchconstituents, a filter may be incorporated into gas metering means 1121and gas directing means 1122.

In another alternative embodiment, as shown in FIGS. 12 a, 12 b, 12 c,12 d, and 12 e, high pressure gas may be supplied to chambers 1218 via acombustion, or other chemical, process conducted within a projectile1200, by pre-charging a chamber 1218 within the projectile 1200 beforeor during launch, or by any other means known in the art for producing acompressed volume of fluid within chamber 1218.

As shown in FIG. 12 a, an exemplary method for pre-charging the chamber1218 before launch involves loading the chamber 1218 with compressedfluid and closing the supply port 1211 as soon as the chamber 1218 isloaded. Accordingly, as described above, the guidance system maymanipulate the direction and/or velocity of the gas exiting theprojectile through the exit ports 1231 to effectively guide theprojectile onto a target. As shown in FIGS. 12 b, 12 c, 12 d, and 12 e,in accordance with other exemplary embodiments, high pressure gas may besupplied to chamber 1218 via a combustion, or other chemical, processconducted within the projectile 1200, or by any other means known in theart for producing a compressed volume of fluid. For example, as shown inFIG. 12 b, compressed gas is supplied to chamber 1218 through chemicalreaction, e.g., combustion, of mono- or bi-propellants. Similarly, FIGS.12 c, 12 d, and 12 e show exemplary embodiments where compressed gas issupplied to chamber 1218, through chemical reaction, e.g., combustion,of solid propellants, liquid propellants, and solid/liquid propellanthybrids.

The selection and/or design of a flight control and ballistic datasystem represented by card or module 26 is believed to be a designchoice and is not the subject of this invention disclosure.Nevertheless, if the projectile is engaged in periodic motion, e.g.,rolling, the present invention provides that trajectory corrections maybe made by waiting for an orientation of a control surface or otherdivert propulsion means to correspond with a direction in which it isdesirable to steer the projectile.

The present invention also contemplates that if the trajectorycorrections are large and the projectile guidance section is rollstabilized, then the projectile can be de-spun and the corrections madewhile the projectile is in a roll-stabilized mode. Then, aftercompleting the corrections, the projectile can go back to being arolling airframe. The present invention enables these capabilities byproviding a mechanism for determining the roll orientation of theprojectile. For example, with reference to FIG. 3 c, the rollorientation of the projectile may be determined through: 1) a roll gyro,2) the roll position of GPS satellites as they turn on/off, 3) GPSattitude determination with multi antennas, and 4) a magnetic turnscounter.

In accordance with an exemplary embodiment, GPS signal data may beacquired from GPS signals transmitted from one or more satellites 78, 79and received by one or more antennas 88, 89 on the projectile. Withfurther reference to FIG. 3 c, in an exemplary embodiment, a pluralityof GPS antenna patterns 88, 89 receive GPS signals which are passedthrough RF processing means 98, 99 to produce RF processed GPS signal3344. Information from a high speed correlator 3354, and roll reference3364 are transmitted to processor 3375 along with RF processed GPSsignal 3344. Based on signals 3344, 3354, and 3364, processor 3374produces a position signal 3381, a velocity signal 3382, and an attitudesignal 3383.

As shown FIG. 3 d, each GPS signal 3407 may be sent through a low noiseamplifier 3417 before being received by radio 3427. Also, in anexemplary embodiment, one or more GPS signal jamming detectors 3418 maybe included to monitor each GPS signal 3407 and to detect whether eachGPS signal has been subjected to jamming. If such jamming is detected,an antijammer 3419 may implement GPS signal protection measures whichmay, for example, selectively and periodically avoid GPS signals thatexhibit jamming based on the periodic motion of the antenna portion ofthe projectile. As shown in FIG. 3 d, jamming detector 3418 andanti-jammer 3419 transmit signals to system processor 3457, which alsoreceives information from roll reference 3467 and GPS receiver 3437.Based on signals received from system processor 3457, digital filters3428 receive GPS signals from S/W radio 3427 and selectively transmitunjammed portions of those GPS signals to GPS receiver 3437. GPSreceiver 3437 then uses pre-stored data 3447 to process and transmit GPSsignals to system processor 3457. Finally, system processor 3457utilizes roll reference 3467 and GPS signal to transmit signals toautopilot 3477, which causes control surfaces and/or divert propulsionmeans 3487 to be actuated, thereby guiding the projectile to a target.

In an exemplary embodiment, a GPS signal protection measure involvescausing the projectile to engage, or remain engaged, in a periodicmotion (e.g., spinning, rolling, or otherwise oscillating) and using theperiodic motion of the projectile to selectively sample GPS signals insuch a manner as to detect and omit jammed GPS signals and to enablereliable processing of unjammed signals. It should be noted that as aprojectile proceeds through each cycle of its periodic motion (i.e.,through each rolling or spinning revolution, oscillation, or otherperiodic motion) each of the one or more antennas 88, 89 willperiodically be positioned to receive a GPS signal from each of the oneor more satellites 78, 79. Thus, so long as at least one GPS signal froma satellite remains capable of being received in an unjammed state byone or more antenna, each of the one or more antennas will periodicallybe free from jamming, e.g., during at least a portion of the rollattitude.

FIG. 8 illustrates three exemplary time-dependent GPS signals asreceived by three corresponding GPS antenna and receiver meansimplemented on a projectile undergoing periodic motion, e.g., spinning.As shown in FIG. 8, a first GPS signal exhibits jamming as theprojectile rotates through an orientation from approximately 30 degreesto an orientation of approximately 210 degrees. Similarly, a second GPSsignal exhibits jamming as the projectile rotates through an orientationfrom approximately 120 degrees to an orientation of approximately 360degrees. Finally, a third GPS signal exhibits jamming as the projectilerotates through an orientation from approximately 360 degrees to anorientation of approximately 30 degrees. Accordingly, a substantiallycontinuous unjammed GPS signal may be received by switching on the firstantenna as the projectile rotates between approximately 210 degrees andapproximately 30 degrees, by switching on the second antenna as theprojectile rotates between approximately 360 degrees and approximately120 degrees, by switching on the third antenna as the projectile rotatesbetween approximately 30 degrees and approximately 360 degrees, byswitching off the first antenna as the projectile rotates betweenapproximately 30 degrees and approximately 210 degrees, by switching offthe second antenna as the projectile rotates between approximately 120degrees and approximately 360 degrees, and by switching off the thirdantenna as the projectile rotates between approximately 360 degrees andapproximately 30 degrees.

With further reference to FIG. 1, in one embodiment, the flight controland ballistic data system 26 is pre-programmed to use informationprovided via bus 32 from the projectile guidance system without gyros12, after launch, to de-spin the projectile and to roughly position thetop of the projectile skyward, or to otherwise orient an antenna pattern88, 89, so as to optimize the reception of GPS signals. In accordancewith this embodiment, after the projectile is de-spun, theaccelerometers are un-caged and activated. The computer, memory andprogram means then performs its first estimate and update of the statematrix including an estimate of the vehicle rates. In accordance withthis exemplary embodiment, where the projectile motion is such that itmay permit one or more antenna 88, 89 to remain oriented so as tocontinuously receive an unjammed GPS signal, the anti-jammer 3419 maydetect that a GPS signal 3407 received by one or more of the antennas isfree from jamming and may switch or otherwise cause the one or more GPSreceiver 18 to use only the unjammed signal. It should be noted that insuch an exemplary embodiment, an unjammed GPS signal may be receivedwithout requiring the projectile to be rolled. To accomplish this, acomputer-implemented process, i.e., a guidance program, may beconfigured to cause the projectile to maintain an orientation of the GPSantenna and receiver means so as to permit one or more antenna tosubstantially continuously receive an unjammed GPS signal.

It should be noted that in such an exemplary embodiment, the projectileneed not be spinning or rolling. Accordingly, this protection measuremay be implemented with projectiles that are not roll-stabilized. Itshould also be noted that, in accordance with this embodiment, theantenna(s) may be decoupled from the main body of the projectile via amechanism such as a slip ring. Accordingly, the projectile may bespinning or roll stabilized while the antennas are maintained in a fixedorientation with respect to one or more satellites. Similarly, theprojectile may be stationary while the antennas undergo periodic motion,e.g., spinning, with respect to one or more satellites.

Thus, in accordance with the invention, the projectile may either bespinning or roll stabilized, e.g., by decoupling it from the main bodyof the projectile via a slip ring. In the event no unjammed GPS signalmay be substantially continuously received, the anti-jamming means maybe configured to cause the projectile to engage in a periodic motion andto selectively and periodically avoid GPS signals that exhibit jamming.

Other exemplary protection measures may include immediate conversion ofthe GPS signal from an RF signal to a digital signal as well asimplementation of an automatic gain control. In such embodiments, eachGPS signal may be converted immediately or very soon after it isreceived by a high speed analog to digital converter. Then, the digitalGPS signal may pass through a digitally implemented automatic gaincontrol circuit. Both the analog to digital converter and the automaticgain control circuit may be implemented in the GPS radio receiver.

In an exemplary embodiment, the system may be configured to beinitialized prior to firing. In accordance with this embodiment, priorto projectile firing, the system may determine and store in memory whichsatellites will be in the field-of-view (FOV) of each antenna pattern88, 89 throughout the trajectory of the projectile and which combinationof satellites 78, 79 gives the best GDOP for the trajectory. Thisinformation is pre-stored in the receiver 18 along with any othercorrection data that may be available.

In an exemplary embodiment, as shown in FIG. 3 d, the GPS signals 3407are transmitted through a low noise amplifier 3417 to a radio receiver3427, which may comprise a high speed analog to digital converter (e.g.,a 5 gigabit/second analog to digital converter with 14 bits ofresolution). Radio receiver 3427 may also comprise a digitallyimplemented automatic gain control circuit. The signals are then sent toa set of digital filters 3427 as well as one or more jamming detector3418. It should be noted that a separate filter 3427 may be implementedfor each pre-selected satellite 78, 79 that will be used in the TOAcomputation. It should also be noted that digital filters 3427 may bedynamically tuned in real time to band pass the frequency of eachpre-selected satellite. The center of the band pass will be the GPScarrier frequency adjusted for Doppler motion between the projectile andthe satellite. The GPS signals are then sent to the GPS receiver 3437where they are correlated against the known code of each of thepre-selected satellites via a high speed digital correlator 3447, andposition and velocity determined.

If the vehicle is rolling, then rate information is extracted byseparating the accelerometer ac signal from the dc or steady statesignal. As the projectile rolls, the pitch and yaw accelerometers outputsignals will vary as a sine wave function. If the vehicle is notrolling, then data is extracted and transferred from the body coordinatesystem to the earth referenced navigation coordinate system using DCM(direction cosine matrix) relationships as required.

FIG. 2 shows a mobile artillery weapon, firing a projectile toward atarget. In the equation shown, {right arrow over (X)}(t)_(True) is avector quantity representing the computed true position of theprojectile as a function of time. {right arrow over (X)}(t)_(Nominal) isa vector representing the ideal location of the projectile on the idealtrajectory. {right arrow over (X)}(0)_(Error) is a vector representingthe position error of the projectile at time zero before launch. {rightarrow over (X)}(t)_(Error) is a vector representing the error in theprojectile's position due to accumulated errors resulting fromnavigational instrument error and GPS errors. The figure shows thesequence of events and schematically portrays an error budget region indashed lines that concludes with an ellipse representing the TARGETERROR that can result from launching the projectile using conventionalartillery practice with no on-board guidance. The funnel region in acontinuous line schematically shows the position error of an on-boardGPS progressively diminishing and asymptotically approaching a limitthat is much smaller than the TARGET ERROR limit for a trajectory withno on-board guidance. The invention projectile guidance system withoutgyros 10 uses a triad of accelerometers to provide real time bodyreferenced data to a computer, memory and program means and real timeposition data from a GPS to establish and output the attitude of theprojectile. The outputs are Kalman filtered and coupled to the flightcontrol and ballistic data system represented by module 26 shown in FIG.1 along with the aiding GPS position data for use in steering theprojectile into the region bounded by the funnel region.

FIGS. 3 a and 3 b schematically show an embodiment of a data flow usedby the projectile guidance system without gyros 10. Phantom block 300represents the function provided by a fire control and ballisticscomputer as it executes an initialization program to process datareceived from data sources such as an external input of a TARGETLOCATION, the geographic location of the weapon launching theprojectile, the PROJECTILE TYPE, the PITCH ALIGNMENT IN BARREL, thepitch angle of the launching tube, the outputs of the triax 304 of threecomponents of acceleration a_(x), a_(y) and a_(z) in body coordinates,the roll angle ROLL φ, and the roll rate ROLL RATE φ from the spincounter 308. The FIRE CONTROL AND BALLISTICS computer processes the dataavailable and prior to launch transfers the initial state data such theinitial orientation angles, body fixed angle rates, present position andinitial velocity components. The spin counter 308 and the triax 304provide their outputs directly to the INITIAL PROJECTILE STATE ESTIMATE312 via signal paths 314 and 333 respectively. It is most likely thatall initial conditions would be input into the projectile from anexternal source such as the FIRE CONTROL AND BALLISTICS function 302 viaan electrical, magnetic or optical link, or perhaps mechanical link. Ifthe gun were to be mounted on a ship, the initial conditions would haveto be extended to include data such as the ship's velocity, heading,displacement from the ship's inertial guidance computer, and possiblythe coordinates of the gun with respect to the station from which thegun is receiving its initial condition information.

The FIRE CONTROL AND BALLISTICS function 302 also provides roll indexinginformation to the spin counter 308 to provide a zero reference. Thespin counter 308 provides estimates of roll angle φ_(b) to an accuracyof a few degrees, as well as roll rate φ_(b). The FIRE CONTROL ANDBALLISTICS function 302 precomputes an expected ballistic trajectorywhich is also passed to the projectile. This ballistic information,along with accelerometer information, or data directly from the FIRECONTROL AND BALLISTICS function 302 is used to determine the initialstate vector and attitudes of the projectile at initialization of theestimation process. The algorithm is initialized when both theaccelerometer and the GPS data are available for state estimation. TheFIRE CONTROL AND BALLISTICS function 302 also receives a set of initialstates for orientation and position of the projectile via signal path320 from the ESTIMATION OF NEW INITIAL TARGET STATE function 322 as aresult of a failure of a fit test.

The FIRE CONTROL AND BALLISTICS function 302 outputs a set of initialSTATE ESTIMATES to the INITIAL PROJECTILE STATE ESTIMATE functionalblock 312 which transfers initial BODY STATE ESTIMATES {right arrow over(V)}₀, {right arrow over (x)}₀ via signal path 324 to SMOOTHING PROCESSfunction block 344 and a set of ATTITUDE STATE ESTIMATES via signal path330 to the initial state estimate of the initial attitude angle array inATTITUDE FILE 332.

The turns or SPIN COUNTER 308 which provided turn count data to the FIRECONTROL AND BALLISTICS function 302 also provides turn count data viasignal line 333 to the INITIAL PROJECTILE STATE ESTIMATE FUNCTION block312 and to the time indexed SPIN COUNTER MEASURED DATA memory arrayrepresented on FIG. 3 b by functional block 350 where the total rollangle φ_((n)) of the projectile is accumulated and time indexed andtransferred via path 351 to the ATTITUDE ESTIMATION ALGORITHM 372. Spincounter technology is a mature technology. The Alliant TechsystemsCompany has reportedly developed one small enough to fit into a 20 mmprojectile. A turns counter for larger diameter projectiles is thereforea product that can be purchased for use in the invention system andprocessed without difficulty. The output of the turns counter is a sinewave output as the projectile rotates through the Earth's MagneticField.

The INITIAL PROJECTILE STATE ESTIMATE FUNCTION block 312 providesinitial ATTITUDE STATE ESTIMATE values for azimuth ψ₀, pitch θ₀ and rollφ₀ angle to the ATTITUDE FILE 332 in FIG. 3 b as initial values.

The outputs of the triax 304 of three components of acceleration a_(x)a_(y) and a_(z) in body coordinates are processed by the functionrepresented by the block 336 to transfer time indexed body referencedchanges in velocity via signal path 338 to the ACCEL DATA FILE memoryarray 340. The ACCEL DATA FILE memory array 340 couples an array of timeindexed acceleration data elements to the SMOOTHING PROCESS functionblock 344.

The filtering and smoothing process of block 344 will be discussed inconnection with FIGS. 6, 7 a and 7 b later in this disclosure. Thesmoothing process uses the initial estimates of attitude, velocity andposition in conjunction with the present accelerometer data to determinevelocity and position of the projectile in body coordinates. The outputsof the smoothing process are STATE VECTOR ESTIMATES, {right arrow over(X)}_(B(n)), {right arrow over (V)}_(B(n)) for position and velocitywhich are coupled via signal path 346 to the BODY TO INERTIALTRANSFORMATION functional block 348.

The GPS MEASURED DATA function block 356 periodically receives timeindexed velocity and position data {right arrow over (V)}_((n)) and{right arrow over (X)}_((n)) and transfers the data via signal path 358into the time indexed array in the computer, memory and program meansmemory shown as function block 360. The GPS velocity and position datais transferred via signal path 362 to a Kalman filter represented byfunction block 364. The Kalman filter transfers filtered values of theprojectile's estimated present position and velocity vectors via signalpath 368 to the FIT ERROR TEST functional block 370 and to the ATTITUDEESTIMATION ALGORITHM functional block 372. The ATTITUDE FILE 332transfers time indexed estimated attitude angles and attitude angularrate data to the ATTITUDE ESTIMATION ALGORITHM functional block 372 viasignal path 374 and to the BALLISTICS ESTIMATE (n) functional block 376via transfer path 378 in earth referenced navigational coordinates. TheBALLISTIC ESTIMATES uses the known model characteristics of theprojectile with the corrected initial position and initial velocityinformation via path 375 and the history of all past attitude andattitude rates from path 378 to provide a modeled estimate of thepresent position and velocity via path 382 to the ATTITUDE ESTIMATIONALGORITHM 372.

The filtered body STATE VECTOR ESTIMATES {right arrow over({circumflexover (X)})}_(B(n)), {right arrow over({circumflex over (V)})}_(B(n))vectors in the orthogonal body coordinate system, are transferred viasignal path 346 to the BODY TO INERTIAL TRANSFORMATION block 348. TheBODY TO INERTIAL TRANSFORMATION block 348 receives the estimatedattitude angles and attitude angular via signal paths 384 and 386 andprocesses the STATE VECTOR ESTIMATES from representation in theorthogonal body coordinate system into vectors in the earth referencednavigational coordinate system to position and velocity estimates inearth referenced navigational coordinates using conventional directioncosine transformations such as those explained in the text by GeorgeSiouris, titled “Aerospace Avionics Systems, A Modern Synthesis”,published by Academic Press, published in 1993. This estimation processcan be simplified and improved using measured turns counter data and bycomputing a new ballistic trajectory based on new initial stateinformation.

The BODY TO INERTIAL TRANSFORMATION functional block 348 outputs theestimated position and velocity STATE VECTOR ESTIMATES in earthreferenced navigational coordinates via signal path 388 to the FIT ERRORTEST functional block 370 where the STATE VECTOR ESTIMATES are comparedto the Kalman filtered GPS position and velocity. It is not possible ina statistical sense to make the estimation process fit better than theGPS position and velocity and the errors associated with the GPS. If theerror for the differences between the two state vectors is less than 1or 2 times the GPS error, the algorithm declares the fit to be“statistically good enough” to quit and wait for a new measured datapoint to arrive. If the test passes, the estimates of position,velocity, attitude and attitude rates are used to update the n−1position in the estimated position and attitude time indexed memoryarrays.

If the test fails, the algorithm estimates a new set of initialprojectile states via a perturbation algorithm and repeats theestimation process again, until the solution converges to limits withinthe predetermined fit error criteria. If the test fails, a FAIL signalis coupled via path 390 to the ESTIMATION OF NEW INITIAL TARGET STATEfunctional block 322 which adjusts the initial position, velocity,attitude and attitude rates and couples a modified set of estimatedstate data via signal path 320 into the INITIAL PROJECTILE STATEESTIMATE functional block 312 for an additional cycle. Iterationcontinues until the initial conditions are adjusted to obtain a PASSsignal out of functional block 370 followed by the next data iterationusing GPS data one second later.

As discussed previously, in order to avoid excessive reliance on GPS andprotection of the GPS signal, higher performing inertial instruments(e.g., inertial instruments performing at levels better than 100 microg) may be used. Honeywell's VBA accelerometers and Silicon Accelerometersupplied from the Litton Guidance & Control Systems Div. of NorthropGrumman Inc. at Woodland Hills, Calif. 91637, the assignee, have such acapability.

A GPS receiver by Alliant Techsystems is available for use in theinvention system and process. The Alliant GPS uses a type 509 processorthat has sufficient processing power and a separate memory which can beused to store user software and the system processor for thisapplication. The Alliant receiver is configured with multiple RF frontends to accommodate multiple antenna inputs as a means of protecting theGPS signal as the projectile rolls. The GPS 509 processor can also beused to process the data and integrate the inertial data with the GPSand turns counter data. Software can be incorporated into the 509processor to extract rate information from the accelerometer triax andintegrate the data from the sensors.

Alternatively, where cost must be reduced, much lower accuracyaccelerometers can be used with a greater reliance on GPS signal dataafter launch. As described above, in an exemplary embodiment, GPS signaldata may be acquired from GPS signals transmitted from one or moresatellites 78, 79 and received by one or more antennas 88, 89 on theprojectile. One or more GPS signal jamming detector 3418 may be includedto monitor each GPS signal 3407 and to detect whether each GPS signal3407 has been subjected to jamming. If such jamming is detected, ananti-jammer 3419 may implement GPS signal protection measures present insystem processor 3457 as shown in FIG. 3 d.

For example, anti-jammer 3419 may cause system processor 3457 to causethe projectile to engage, or remain engaged, in a periodic motion suchas rolling or spinning. As the projectile proceeds through each cycle ofthe periodic motion, i.e., through each rolling revolution, each of theone or more antennas 88, 89 may periodically be positioned to receive aGPS signal from each of the one or more satellites 78, 79. Thus, so longas at least one GPS signal from a satellite remains capable of being byone or more antennas in an unjammed state, each of the one or moreantennas will periodically be free from jamming, e.g., during at least aportion of the roll attitude.

Alternatively, where the jamming detector 3418 does not detect jamming,or detects the substantial absence of jamming, in one or more GPSsignal, antijammer 3419 may cause system processor 3457 to de-spin theprojectile and to roughly position the top of the projectile skyward, orto otherwise orient an antenna pattern 88, 89, so as to optimize thereception of GPS signals 3407. In accordance with this embodiment, afterthe projectile is de-spun, the accelerometers are un-caged andactivated. The computer, memory and program means then performs itsfirst estimate and update of the state matrix including an estimate ofthe vehicle rates. In accordance with this exemplary embodiment, theantijammer 3419 may switch or otherwise cause the one or more GPSreceiver 18 or the system processor 3457 to use only the unjammed GPSsignal received by the one or more of the antennas that, is free fromjamming.

The roll angle of a guided projectile is typically provided by a gyro ora gyro platform. A roll angle signal characterizing the roll angle ofthe projectile is necessary to enable a flight control system to drivethe flight control surfaces, such as canards, to roll the projectile tothe left or right as required, to use lift from its lift surfaces tochange the heading of the projectile to one directed to a target. It isan objective of this disclosure to provide a mathematical and therefore,a computer programmable process, for generating the roll angle of theprojectile without signals from a gyro and by using the accelerometeroutputs of a triax and the position and velocity signals from a GPS.

This technical description uses the following notation. {overscore (x)}denotes a vector with no specific reference frame of resolution.{overscore (x)}^(b) denotes a vector resolved in a coordinate framecalled the body-frame or projectile-frame. All coordinate framesdiscussed herein, including the body-frame coordinates, are right-handedorthogonal frames that have x, y, and z axes that extend from a locationin the body of the projectile, such as the center of gravity, or origindesignated as a reference point “O” to form an orthogonal triadextending in the forward or positive “x_(b)” axis direction, in theright or positive “y_(b)” axis direction and down in the positive“z_(b)” axis direction. At least two accelerometers are fixed to andaligned with the body-frame along the y_(b) and z_(b) axes. Theprincipal axes of the b-frame coincide with the input axes of theinertial sensors and intersect at the origin “O”. A second coordinateframe of interest is the geographic or earth referenced navigationalcoordinate system or frame with principal axes X, Y and Z that coincidewith the East, North, and Up directions as shown at the bottom of FIG.4.

Subscripts on vectors are used to indicate a particular property oridentification of the vector. Matrices are designated with capitalletters. C_(N) ^(b) denotes a direction cosine matrix (DCM) thattransforms a vector from the N-frame or Navigation-frame to thebody-frame, i.e., {right arrow over (x)}^(b)=C_(N) ^(b){right arrow over(X)}^(N). Time dependency of a quantity is indicated with round bracketsaround a time variable or index. For example, C_(N) ^(b) (t₁) denotesthe value of the DCM for the conversion from navigational to body-framecoordinates at time t₁. The transpose of the C_(N) ^(b) (t₁) DCMprovides C_(b) ^(N) (t₁), the DCM for the conversion from body-frame tonavigational-frame coordinates at time t₁.

An increment of a variable is indicated with the symbol Δ. For example,Δ{overscore (x)} denotes the increment of the vector {overscore (x)}over a predefined time interval. An error in a variable is indicatedwith the symbol δ. For example, δ{overscore (x)} denotes the error inthe vector {overscore (x)}. The=symbol indicates an approximate orsubstantially close equality.

Sensors and Orientation

In a two accelerometer arrangement, a first accelerometer is positionedto sense acceleration along the y_(b) body axis that extends outwardfrom the right side of the body. A second accelerometer is positioned tosense acceleration along the z_(b) body axis that extends downward fromthe origin forming the yaw axis. A third accelerometer, not required forthe computation of a roll variable, is positioned to sense accelerationalong the x_(b) or forward axis.

Roll Angle Computation

The accelerometer outputs are resolved from body coordinates intonavigational coordinate values in accordance with the followingequation:A ^(N) =C _(b) ^(N) A ^(b) ={T}{H} ^(T) {P} ^(T) {R} ^(T) A ^(b)  1a.where A^(b) represents the acceleration vector in body coordinates, thebracketed H, P and R operators {H}^(T), {P}^(T) {R}^(T) individuallyrepresent the transpose of the positive rotation DCM transformationsteps from body to navigational coordinates for roll, about x_(b),pitch, about y_(b) and heading or yaw, about z_(b), of the vehicle orprojectile fixed coordinate system, in that sequence, to obtain theacceleration vector A^(N) as elements of vehicle acceleration innavigation coordinates with components along the fixed navigation orearth referenced East, X, North, Y and Up, Z axis.

DCM {T} represents a unitary matrix that transforms the body referencedacceleration data from a North, East, Down system into an East, North,Up frame and is defined as: $\begin{matrix}{\{ T \}\overset{\Delta}{=}{\quad\begin{matrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & {- 1}\end{matrix}\quad }} & {1{b.}}\end{matrix}$

The data elements of acceleration A^(b), from the triad ofaccelerometers on the projectile, are integrated over a predeterminedincremental time interval such as one second to obtain incrementalchanges in velocity in navigational coordinates in accordance with thefollowing equation:ΔV ^(N) =∫C _(b) ^(N) A ^(b) dt={T}{H} ^(T) {P} ^(T) {R} ^(T) A ^(b)ΔT  2.

Concurrent with each calculation of the incremental change in velocitybased on accelerometer data, GPS data is used to calculate anincremental change in velocity. The GPS data is already in navigationalcoordinates. The following equation is used:ΔV _(GPS) =V _(GPS2) −V _(GPS1) =ΔV ^(N) ={T}{H} ^(T) {P} ^(T) {R} ^(T)A ^(b) ΔT  3.

A double integration is performed on the body accelerometer data alongwith a body to navigation frame transformation to obtain the change inposition based on transformed accelerometer data from the body frameusing the following equation:ΔP ^(N) =∫∫C _(b) ^(N) A ^(b) dtdτ={T}{H} ^(T) {P} ^(T) {R} ^(T) A ^(b)ΔT ²  4.

With each calculation of the incremental change in position based onaccelerometer data, ΔP^(N), GPS data is used to calculate asubstantially equal GPS incremental change in position, ΔP_(GPS). Thefollowing equation is used:ΔP _(GPS) =P _(GPS2) −P _(GPS1) =ΔP ^(N) ={T}{H} ^(T) {P} ^(T) {R} ^(T)A ^(b) ΔT ²  5.

Group Equations

The product of a DCM matrix and its transpose is equal to the identitymatrix. Therefore{P} ^(T) {P}={I}  6a. {H} ^(T) {H}={I}  6b.

From equation 3, it is possible to relate change in velocity componentsor accelerations components from the GPS data to acceleration componentsfrom accelerometer data as follows:ΔV _(GPS) ={T}{H} ^(T) {P} ^(T) {R} ^(T) A ^(b) ΔT=ΔV ^(N)  7a.

From equation 5, it is possible to relate change in position componentsor velocity components from the GPS data to velocity components fromaccelerometer data as follows:ΔP _(GPS) ={T}{H} ^(T) {P} ^(T) {R} ^(T) A ^(b) ΔT ² =ΔP ^(N)  7b.

If the pitch, heading and unitary transformation DCMs are grouped ormultiplied to form a single DCM, {Q}, and the increment in time ischosen for simplicity to be 1 second (however, it should be understoodthat any time increment can be used in the equations):{Q}={P}{H}{T}  8a.ΔT=1 Second and Δ²=1 Second squared=1  8b.

Substituting 1 for ΔT in equation 7a and 1 for ΔT² in equation 7b andmultiplying both sides of equations 7a and 7b by the transpose of theunitary, heading and pitch DCMs, in that order, provides the followingequations:{P}{H}{T}(ΔV _(GPS))={R}A ^(b) ={Q}(ΔV _(GPS))  9a.{P}{H}{T}(ΔP _(GPS))={R}A ^(b) ={Q}(ΔP _(GPS))  9b.

Using CP to represent Cos, θ SP to represent Sin θ, CH to represent Cosψ, SH to represent Sin ψ, CR to represent Cos φ and SR to represent Sinφ, then the six DCMs for the rotation conversions from body tonavigation coordinates are: $\begin{matrix}\begin{matrix}{\{ P \}\overset{\Delta}{=}{\quad\begin{matrix}{CP} & 0 & {- {SP}} \\0 & 1 & 0 \\{SP} & 0 & {CP}\end{matrix}\quad }} & {\{ P \}^{T}\overset{\Delta}{=}{\quad\begin{matrix}{CP} & 0 & {SP} \\0 & 1 & 0 \\{- {SP}} & 0 & {CP}\end{matrix}\quad }}\end{matrix} & {10a} \\\begin{matrix}{\{ H \}\overset{\Delta}{=}{\quad\begin{matrix}{CH} & {SH} & 0 \\{- {SH}} & {CH} & 0 \\0 & 0 & 1\end{matrix}\quad }} & {\{ H \}\overset{\Delta}{=}{{\quad\begin{matrix}{CP} & {- {SH}} & 0 \\{SH} & {CH} & 0 \\0 & 0 & 1\end{matrix}\quad }\quad{and}}}\end{matrix} & {10{b.}} \\\begin{matrix}{\{ R \}\overset{\Delta}{=}{\quad\begin{matrix}1 & 0 & 0 \\0 & {CR} & {SR} \\0 & {- {SR}} & {CR}\end{matrix}\quad }} & {\{ P \}\overset{\Delta}{=}{\quad\begin{matrix}1 & 0 & 0 \\0 & {CR} & {- {SR}} \\0 & {SR} & {CR}\end{matrix}\quad }}\end{matrix} & {10{c.}}\end{matrix}$

Where the body is a projectile fired from a gun, the pitch angle is theangle of the barrel at firing and the heading is the angle ψ that thebarrel makes with true north at firing. Both are available frominitialization from the system and are expected to be close to thenormal trajectory with time, and therefore nominally known.

The elements of {Q} are known from 8a, 10a, and 10b and are believed tobe slowly varying, so let {Q} be: $\begin{matrix}{\{ Q \} = {\quad\begin{matrix}q_{11} & q_{12} & q_{13} \\q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad }} & 11.\end{matrix}$

Rewriting equations 9a and 9b and substituting {Q} into the result for{P} {H} {T} with ΔT=1 second and ΔT²=1 second provides,{Q}(ΔV _(GPS))={R}A ^(b)  12a. {Q}(ΔP _(GPS))={R}A ^(b)  12b.

Reversing the left and right sides:{R}A ^(b) ={Q}ΔV _(GPS))  13a.{R}A ^(b) ={Q}(ΔP _(GPS))  13b.

Since the roll DCM is characterized in 10c and accelerometer data is avector: $\begin{matrix}{\{ R \} = {\quad\begin{matrix}1 & 0 & 0 \\0 & {CR} & {SR} \\0 & {- {SR}} & {CR}\end{matrix}\quad }} & {10{d.}}\end{matrix}$  A ^(b) =a ₁ {circumflex over (x)} _(b) +a ₂ ŷ _(b) +a ₃{circumflex over (z)} _(b)  14.

where {circumflex over (x)}_(b), ŷ_(b) and {circumflex over (z)}_(b) areunit vectors along the body axis.

Equations 13a and 13b are rewritten as: $\begin{matrix}{{{\quad\begin{matrix}1 & 0 & 0 \\0 & {CR} & {SR} \\0 & {- {SR}} & {CR}\end{matrix}\quad }{\begin{matrix}a_{1} \\a_{2} \\a_{3}\end{matrix}}} = {{\quad\begin{matrix}q_{11} & q_{12} & q_{13} \\q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad }( \overset{\_}{\Delta\quad V_{GPS}} )}} & {15{a.}} \\{{{\quad\begin{matrix}1 & 0 & 0 \\0 & {CR} & {SR} \\0 & {- {SR}} & {CR}\end{matrix}\quad }{\begin{matrix}a_{1} \\a_{2} \\a_{3}\end{matrix}}} = {{\quad\begin{matrix}q_{11} & q_{12} & q_{13} \\q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad }( \overset{\_}{\Delta\quad P_{GPS}} )}} & {15{b.}}\end{matrix}$

The incremental accelerometer and velocity data from the GPS is madeavailable in fixed navigation coordinates forming two three-elementvectors{overscore (ΔV _(GPS))}=ΔV _(x) {circumflex over (x)}+ΔV _(y) ŷ+ΔV _(z){circumflex over (z)} and  16a.{overscore (ΔP _(GPS))}=ΔP _(x) {circumflex over (x)}+ΔP _(y) ŷ+ΔP _(z){circumflex over (z)}  16b.where, {circumflex over (x)}, ŷ and {circumflex over (z)} are basic unitvectors of the East, North and Up coordinate frame system.

The matrix for {Q} in 15a and 15b are multiplied by the vectors of 16aand 16b as: $\begin{matrix}{{{\quad\begin{matrix}1 & 0 & 0 \\0 & {CR} & {SR} \\0 & {- {SR}} & {CR}\end{matrix}\quad }{\begin{matrix}a_{1} \\a_{2} \\a_{3}\end{matrix}}} = {{\quad\begin{matrix}q_{11} & q_{12} & q_{13} \\q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad }{\begin{matrix}{\Delta\quad V_{x}} \\{\Delta\quad V_{y}} \\{\Delta\quad V_{z}}\end{matrix}}}} & {17{a.}} \\{{{\quad\begin{matrix}1 & 0 & 0 \\0 & {CR} & {SR} \\0 & {- {SR}} & {CR}\end{matrix}\quad }{\begin{matrix}a_{1} \\a_{2} \\a_{3}\end{matrix}}} = {{\quad\begin{matrix}q_{11} & q_{12} & q_{13} \\q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad }{\begin{matrix}{\Delta\quad P_{x}} \\{\Delta\quad P_{y}} \\{\Delta\quad P_{z}}\end{matrix}}}} & {17{b.}}\end{matrix}$

The left sides of 17a and 17b are multiplied out forming a 3×1 matrix onthe left: $\begin{matrix}{\lbrack \quad\begin{matrix}a_{1} \\{{a_{2}{CR}} + {a_{3}{SR}}} \\{{{- a_{2}}{SR}} + {a_{3}{CR}}}\end{matrix}\quad \rbrack = {{\quad\begin{matrix}q_{11} & q_{12} & q_{13} \\q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad }{\begin{matrix}{\Delta\quad V_{X}} \\{\Delta\quad V_{y}} \\{\Delta\quad V_{z}}\end{matrix}}}} & {18{a.}} \\{\lbrack \quad\begin{matrix}a_{1} \\{{a_{2}{CR}} + {a_{3}{SR}}} \\{{{- a_{2}}{SR}} + {a_{3}{CR}}}\end{matrix}\quad \rbrack = {{\quad\begin{matrix}q_{11} & q_{12} & q_{13} \\q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad }{\begin{matrix}{\Delta\quad P_{X}} \\{\Delta\quad P_{y}} \\{\Delta\quad P_{z}}\end{matrix}}}} & {18{b.}}\end{matrix}$

The first element a₁ of the vector on the left-hand side of equations18a and 18b, i.e., has no roll angle component because it representsacceleration down the vehicle's longitudinal axis; therefore it isdeleted leaving only the second and third row expressions. FromEquations 18a and 18b: $\begin{matrix}{\lbrack \quad\begin{matrix}{{a_{2}{CR}} + {a_{3}{SR}}} \\{{{- a_{2}}{SR}} + {a_{3}{CR}}}\end{matrix}\quad \rbrack = {\lbrack \quad\begin{matrix}q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad \rbrack{\begin{matrix}{\Delta\quad V_{X}} \\{\Delta\quad V_{y}} \\{\Delta\quad V_{z}}\end{matrix}}}} & {19{a.}} \\{\lbrack \quad\begin{matrix}{{a_{2}{CR}} + {a_{3}{SR}}} \\{{{- a_{2}}{SR}} + {a_{3}{CR}}}\end{matrix}\quad \rbrack = {\lbrack \quad\begin{matrix}q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad \rbrack{\begin{matrix}{\Delta\quad P_{X}} \\{\Delta\quad P_{y}} \\{\Delta\quad P_{z}}\end{matrix}}}} & {19{a.}}\end{matrix}$

The left side of 19a and 19b can be rewritten to form a 2×2 matrix timesa 2×1 matrix as: $\begin{matrix}{\lbrack \quad\begin{matrix}{{a_{2}{CR}} + {a_{3}{SR}}} \\{{{- a_{2}}{SR}} + {a_{3}{CR}}}\end{matrix}\quad \rbrack = {{\begin{bmatrix}{a_{2}a_{3}} \\{a_{3} - a_{2}}\end{bmatrix}\begin{bmatrix}{CR} \\{CR}\end{bmatrix}} = {\lbrack \quad\begin{matrix}q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad \rbrack{\begin{matrix}{\Delta\quad V_{X}} \\{\Delta\quad V_{y}} \\{\Delta\quad V_{z}}\end{matrix}}}}} & {20{a.}} \\{\lbrack \quad\begin{matrix}{{a_{2}{CR}} + {a_{3}{SR}}} \\{{{- a_{2}}{SR}} + {a_{3}{CR}}}\end{matrix}\quad \rbrack = {{\begin{bmatrix}{a_{2}a_{3}} \\{a_{3} - a_{2}}\end{bmatrix}\begin{bmatrix}{CR} \\{CR}\end{bmatrix}} = {\lbrack \quad\begin{matrix}q_{21} & q_{22} & q_{23} \\q_{31} & q_{32} & q_{33}\end{matrix}\quad \rbrack{\begin{matrix}{\Delta\quad P_{X}} \\{\Delta\quad P_{y}} \\{\Delta\quad P_{z}}\end{matrix}}}}} & {20{b.}}\end{matrix}$

Combining 20a and 2Db obtains: $\begin{matrix}{{\begin{bmatrix}a_{2} & a_{3} \\a_{3} & {- a_{2}} \\a_{2} & a_{3} \\a_{3} & {- a_{2}}\end{bmatrix}\lfloor \begin{matrix}{CR} \\{SR}\end{matrix} \rfloor} = {\lbrack \quad\begin{matrix}q_{21} & q_{22} & q_{23} & 0 & 0 & 0 \\q_{31} & q_{32} & q_{33} & 0 & 0 & 0 \\0 & 0 & 0 & q_{21} & q_{22} & q_{23} \\0 & 0 & 0 & q_{31} & q_{32} & q_{33}\end{matrix}\quad \rbrack\begin{bmatrix}{\Delta\quad V_{X}} \\{\Delta\quad V_{Y}} \\{\Delta\quad V_{Z}} \\{\Delta\quad P_{X}} \\{\Delta\quad P_{Y}} \\{\Delta\quad P_{Z}}\end{bmatrix}}} & {20{c.}}\end{matrix}$

A calculation of the roll angle will use the following definitions:$\begin{matrix}{\alpha_{1} = \begin{bmatrix}a_{2} & a_{3} \\a_{3} & {- a_{2}} \\a_{2} & a_{3} \\a_{3} & {- a_{2}}\end{bmatrix}} & {20{d.}}\end{matrix}$is the accelerometer coefficient matrix. $\begin{matrix}{\beta_{1} = \begin{bmatrix}q_{21} & q_{22} & q_{23} & 0 & 0 & 0 \\q_{31} & q_{32} & q_{33} & 0 & 0 & 0 \\0 & 0 & 0 & q_{21} & q_{22} & q_{23} \\0 & 0 & 0 & q_{31} & q_{32} & q_{33}\end{bmatrix}} & {20{e.}}\end{matrix}$is the pitch/heading coefficient matrix. $\begin{matrix}{\overset{\_}{M_{1}} = \begin{bmatrix}{\Delta\quad V_{X}} \\{\Delta\quad V_{Y}} \\{\Delta\quad V_{Z}} \\{\Delta\quad P_{X}} \\{\Delta\quad P_{Y}} \\{\Delta\quad P_{Z}}\end{bmatrix}} & {20{f.}}\end{matrix}$is the GPS information vector and $\begin{matrix}{{\overset{\_}{\chi}}_{1} = \begin{bmatrix}{CR} \\{SR}\end{bmatrix}} & {20{g.}}\end{matrix}$is the unknown roll quantity. Rewriting Equation 20c provides:α₁{overscore (χ)}₁=β₁ {overscore (M)} ₁  21.

The α₁ matrix is a 4×2 matrix. {overscore (χ)}₁ is a 2×1 vector. SolvingEquation 21 for {overscore (χ)}₁, recognizing that Equation 21represents four equations in two unknowns (CR, SR), proceeds withmultiplying both sides of Equation 21 by α₁ ^(T) to provide:α₁ ^(T)α₁{overscore (χ)}₁=α₁ ^(T)β₁ {overscore (M)} ₁  22a.

Pre-multiplying both sides by the inverse of the product of the α₁ ^(T)α₁ provides:{overscore (χ)}₁=[α₁ ^(T)α₁]⁻¹α₁ ^(T)β₁ {overscore (M ¹ )}  22b.

Therefore, since $\begin{matrix}{{\overset{\_}{\chi}}_{1} = {\begin{bmatrix}{CR} \\{SR}\end{bmatrix} = \begin{bmatrix}{\chi(1)} \\{\chi(2)}\end{bmatrix}}} & {23{a.}}\end{matrix}$the roll angle is:Roll(φ=tan⁻¹(χ₁(2)/χ₁(1)))=tan⁻¹(SR/CR)  23b.

This procedure is followed for additional sample times by again solvingthe over-determined system as follows. For t=t₂ there will be eightequations with two unknowns (cos R, sin R) and for t=t_(n) there will be4n equations with two unknowns (cos R, sin R).

The iterative solution is found by stacking the equations as follows(i.e., for t=t₂): $\begin{matrix}{{\begin{bmatrix}\alpha_{1} \\\alpha_{2}\end{bmatrix}\quad{\overset{\_}{\chi}}_{2}} = \begin{bmatrix}{\beta_{1}\overset{\_}{M_{1}}} \\{\beta_{2}\overset{\_}{M_{2}}}\end{bmatrix}} & 24.\end{matrix}$where α₁, β₁ and {overscore (M)}₁ have been defined above. The newvariables α₂, β₂ and {overscore (M)}₂ have the same structure as α₁, β₁and {overscore (M)}₁ except that they represent the values at t=t₂.Also, {overscore (χ)}₂ represents the values of cos R, sin R utilizingthe variable over two data samples. The solution at t=t₂ is thereforeobtained as before and as follows: $\begin{matrix}{{{\lbrack {\alpha_{1}^{T}\alpha_{2}^{T}} \rbrack\quad\begin{bmatrix}\alpha_{1} \\\alpha_{2}\end{bmatrix}}{\overset{\_}{\chi}}_{2}} = {\lbrack {\alpha_{1}^{T}\alpha_{2}^{T}} \rbrack\begin{bmatrix}{\beta_{1}\overset{\_}{M_{1}}} \\{\beta_{2}\overset{\_}{M_{2}}}\end{bmatrix}}} & 25. \\{{\overset{\_}{\chi}}_{2} = {\lbrack {\sum\limits_{i = 1}^{2}\quad( {\alpha_{i}^{T}\alpha_{i}} )} \rbrack^{- 1}\lbrack {\sum\limits_{i = 1}^{2}\quad{\alpha_{i}^{T}\beta_{i}\overset{\_}{M_{1}}}} \rbrack}} & 26.\end{matrix}$

Similarly at t=t_(n), for n data samples, $\begin{matrix}{{\overset{\_}{\chi}}_{n} = {\lbrack {\sum\limits_{i = 1}^{n}\quad( {\alpha_{i}^{T}\alpha_{i}} )} \rbrack^{- 1}\lbrack {\sum\limits_{i = 1}^{n}\quad{\alpha_{i}^{T}\beta_{i}\overset{\_}{M_{i}}}} \rbrack}} & 27. \\{{Roll}_{n} = {\tan^{- 1}( \frac{\chi_{n}(2)}{\chi_{n}(1)} )}} & 28.\end{matrix}$

For a generalized projectile or vehicle we also need data for the pitchand heading coefficient matrix. The heading and pitch (no gyro) might beapproximated from the ground track (arctan Vn/Ve), and the pitch fromthe flight path angle (arctan of the vertical velocity or rate of changeof altitude and the ground speed as determined from delta P North anddelta compass available from Honeywell. If the winds aloft are known,they might be used to correct the heading and if the angle of attackwere known, it might be used to correct the pitch angle.

FIG. 5 is a block diagram FIG. 5 is a block diagram that schematicallyprovides an overview of the process functions necessary for roll anglecomputation. GPS RECEIVER, block 500 is shown with its internal computerproviding POSITION and VELOCITY data via POSITION sampler 502, andVELOCITY SAMPLER 504. Each sampler has a respective sampler delay 506and 508. The outputs of the samplers are Δ POSITION and Δ VELOCITY andthis data is coupled to a ROLL ESTIMATE (Iterative Solution) functionalblock 510. The ACCELEROMETERS block 512 represents the triax ofaccelerometers aligned with the BODY AXES described earlier. Thecomponent of acceleration a_(x) is aligned with the longitudinal axis ofthe projectile as shown in FIG. 1. The component of acceleration a_(x)changes very slowly during the flight from one second to the next. Thecomponent of acceleration a_(y) is the cross track acceleration and thea_(z) is the component of acceleration along the up axis both beingshown in FIG. 1. The PITCH HEADING COEFFICIENT MATRIX 514 is initializedwith a_(y) and a_(z) accelerometer data. Processing of the data usespresent PITCH ESTIMATE and HEADING ESTIMATE for trigonometric values.The input ΔT will be shown to have the preferred value of one second.The PITCH HEADING COEFFICIENT MATRIX is used with the GPS Δ POSITION andΔ VELOCITY inputs by the ROLL ESTIMATE functional block 510 to provide aroll angle φ output to the FLIGHT CONTROL block 516 which includes block26 in FIG. 1.

Smoothing

The smoothing process referred to by block 344 in FIG. 3 a will now beexplained in connection with a discussion of FIGS. 6, 7 a and 7 b. Theinvention projectile guidance system with accelerometers and a GPSreceiver outputs a smoothed estimate of the past trajectory of theprojectile from launch to its present position, throughout its flight,to the FLIGHT CONTROL block 26, shown on FIG. 1, along with the attitudeangles to permit the flight control function 26 to adjust the heading orground track of the projectile to the known target location.

FIG. 6 provides an embodiment of a forward smoother. The embodimentshown uses Kalman filter. The Kalman filter of block 364, on FIG. 3 b issuch a filter in which there have been n measurements {y₁, y₂, y₂, . . ., y_(n)} at times t₁t₂, . . . , t_(n).

A measurement at time t_(k) might contain acceleration data from aninstrument or a velocity or a piece of position data or a sample ofvelocity or position data from a GPS input or from accelerometer data asdelta V data and these values are arrayed as values of the y vector as{y₁, y₂, y₂, . . . , y_(n)}. We want an estimate of the system state ateach update.

The process is time indexed or sample indexed with GPS information at aone Hz sample rate. Between GPS samples, the system reprocesses all pastsamples and states using the measurement index counter or k counter toprovide an index as it counts from 1 to n.

The process has the objective of going back in time to obtain aneducated estimate of the system state vector at an earlier sample timet_(k). Measurement data, such as GPS position data, is indexed as it isacquired or received as samples 1 through k with the last samplereceived being measurements indexed as sample n.

FIG. 6 starts at block 600 with the initialization of the covariancematrix P(0)=P(t₀) at t=t₀ based on a knowledge of the initial state ofthe process at startup. Block 602 shows that the sample index counter,the k counter, is then set to one and block 604 tests to see if time tis greater than t_(n),. If time is less than or equal to t_(n), the testadvances to block 606 and increments the index counter by one.

Block 608 represents the step of propagating the system state vector orestimated state vector x from time t_(k−1) to time t_(k). The symbolφ_(k−1) represents the system equation or transfer function. Thetransfer function φ_(k−1) propagates the state vector of the system xfrom time k−1 to time k. The state vector expressed as a state matrix“x” is a column matrix that is “m” by one in dimension. The covariancematrix P is “m” by “m” in dimension.

The processes advances to block 610. In this step, the flow chartpropagates the covariance matrix P from t_(k−1) to t_(k), using theequation P_(k) ^({overscore ( )})=φ_(k−1)P_(k−1)φ_(k−1) ^(T)Q_(k−1) Inthis equation, the term P_(k) is the error covariance associated withthe filter estimate of the state vector {circumflex over (x)}_(k)^({overscore ( )}). The transfer function φ_(k−1) and its transposeφ_(k−1) ^(T) are introduced above. The term Q_(k−1) is the covariancematrix of the process noise.

In block 612, the process computes the Kalman filter gain for timet=t_(k). The symbol G_(k) represents the Kalman gain. The term H_(k) isthe measurement or observation matrix at time t=t_(k), and H_(k) ^(T)represents the transpose of the measurement matrix at time t=t_(k). Themeasurement matrix will typically be formed from elements such as Vx,Vy. The y vector or measurement vector is formed from the product of theH matrix and the state vector x, the product being added to themeasurement noise matrix.

Within the brackets of the expression in block 612, the error covariancematrix P_(k) ^({overscore ( )}) is multiplied by the transpose of themeasurement matrix H_(k) ^(T). The product of the two is then multipliedby the measurement matrix H_(k). The result of the product is added toR_(k), the covariance matrix of the measurement noise vector.

The R_(k) matrix is obtained from information that is gathered by thesystems engineers through empirical testing. The systems engineersdetermine what noise is associated with each sensor by collectinghistorical noise data on each measurement variable. The H_(k) matrix isgiven and is hardware and instrument dependent. The Kalman gain isdetermined and then used to update the estimate of the state vectors inblock 614. As each measurement is taken, it is itself a function of acombination of the states. For example, the velocity of a projectilemight be the result of x and y velocity components. The result is alinear combination of the individual measurements.

The process advances to block 614 and the system state estimate vectoris updated for time t_(k) using the equation: {circumflex over(x)}_(k)={circumflex over (x)}_(k)^({overscore ( )})+G_(k)[y_(k)−H_(k){circumflex over (x)}_(k)^({overscore ( )})]. The term H_(k) multiplies the term {circumflex over(x)}_(k) ^({overscore ( )}). The result is subtracted from y_(k). Theresult is an observable difference that is multiplied by the Kalman gainand then added to the previously estimated state vector {circumflex over(x)}_(k) ^({overscore ( )}).

The minus sign in the superscript of {circumflex over (x)}_(k)^({overscore ( )}) implies that it represents the state a little beforethe kth update. The sample index “k” can range in value from 1 to nwhere n is the most recent update index in time. A state variablewithout a superscript implies that the state variable value is a valuethat exists just after an update. The symbol {circumflex over (x)}_(k)represents the kth estimate of the state vector x. When the y_(k)measurement is made, the kth update is performed for the y_(k)measurement. The measurement index “k” is a running index.

After updating the state vector at time t=t_(k) by evaluating theequation in block 614, the process proceeds to the equation in block616:P _(k) =[I−G _(k) H _(k) ]P _(k) ^({overscore ( )})which is used to update the covariance matrix P_(k) The matrix I is anidentity matrix. All of its elements are zero except the main diagonalelements which are ones. The dimension of the identity matrix is matchedto the product of G_(k) and H_(k) matrices. All of the values necessaryfor the computation of the covariance matrix are available from previoussteps.

After the computation of the P_(k) matrix, the process leaves block 616and transfers back to decision block 604 to once again test to see ift>t_(n). As time reaches t_(n), all of the measurements have beenprocessed. No additional measurements are to be made. A “yes” resulttransfers the process to the Scenario Complete block 618 and the missionis completed. If the flight had lasted one hundred seconds and the GPSprovided position measurements at one per second, there would have been100 GPS samples and the sample index k would have been incremented from1 to 100. The index “k” cannot exceed 100.

FIGS. 7 a and 7 b form a fixed interval smoother, FIG. 7 a providing aforward pass and FIG. 7 b providing a backward pass. FIG. 7 a can beseen to be identical in function to that of FIG. 6 with the exception ofblock 700. The process exits block and advances to block 702 also foundat the top of FIG. 7 b. The backward sweep begins with block 704 as theprocess sets the sample index counter, i.e., the k counter to currentindex value of n. The object of the backward sweep process of FIG. 7 bis to compute the smoother estimate x_(s)(k)={circumflex over (x)}(k|n)and the smoother error covariance P_(s)(k)=P(k|n), using all nmeasurements as the measurement index counter “k” is decremented k=n,n−1, n−2, . . . , 1. A subscript “s” in the expression x_(s) (k) and inthe expression P_(s)(k) indicates a smoothed variable.

The process advances to block 706 to compute smoother gain:A(k)=P(k|k)φ_(k) ^(T) P ⁻¹(k+1|k)

The vertical bar followed by a “k” in a term such as (k|k) means thatthe matrix will be evaluated for all measurements up to and includingthe kth measurement. The backward sweep begins only after the conclusionof a forward sweep. All k measurements are available at the conclusionof the forward sweep. All k data points form a fixed interval, and all nmeasurements from t_(k) back to t₁ are used during the backward sweep.

The covariance matrix P(k|k) is multiplied with the transpose of thetransfer function φ_(k) at time t=k.

The rightmost term in block 706 is the inverse of the covariance matrixP(k+1|k) at time t_(k+1) using all k measurements. The vertical barfollowed by “k” shows that all of the data for measurements through k isto be used.

The process then advances to block 708. The object of the equation inblock 708 is to compute a smoothed state vector {circumflex over(x)}(k|n) using all n measurements. The equation in block 708 is:{circumflex over (x)}(k|n)={circumflex over (x)}(k|k)+A(k)[{circumflexover (x)}(k+1|n)−{circumflex over (x)}(k+1|k)]

The gain matrix A(k) is available from block 706. The first term afterthe equal sign is the kth state vector estimate using measurement datathrough measurement k. The first term inside of the bracket, the term{circumflex over (x)}(k+1|n) represents the estimate of the state vectorat time k+1 using data for all measurements through measurement n. Thesecond term inside of the bracket {circumflex over (x)} (k+1|k) is anestimate of the state vector at time k+1 using all measurements up totime t_(k). The difference is calculated as the second term issubtracted from the first. The result is multiplied by the gain matrix,the product is added to {circumflex over (x)}(k|k).

The process then advances to block 710 to compute a smoothed covariancematrix from the equation:P(k|n)=P(k|k)+A(k)[P(k+1|n)−P(k+1|k)]A ^(T)(k)

The first covariance term after the equal sign P(k|k) uses all datathrough the kth update. The first covariance term in the bracketP(k+1|n) uses all of the data from measurements through measurement n.The second covariance term in the bracket P(k+1|k) uses only data upthrough the kth update. A covariance difference is calculated from thecovariance terms within the brackets. The covariance difference term isthen multiplied by the gain matrix and the result is added to P(k|k).

After the smoothed covariance estimate is calculated, the processadvances to block 712 and decrements the measurement index counter fromk to k−1. The process then advances to decision block 714 and a test ismade to determine if the measurement index matrix k is greater than one.Looping or cycling continues with each “no” decision followed by a cycleback to block 706 until k=1. When k=1, the test at block 714 results ina “yes” decision and the process advances to block 716, the BackwardSweep Complete block. Block 716 signals the conclusion of the program asthe projectile nears its target.

Those skilled in the art will appreciate that various adaptations andmodifications of the preferred embodiments can be configured withoutdeparting from the scope and spirit of the invention. Therefore, it isto be understood that the invention may be practiced other than asspecifically described herein, within the scope of the appended claims.

1. A projectile guidance system with accelerometers and a GPS receivercomprising: a projectile having a triax of accelerometers mounted in theprojectile and providing roll, pitch and azimuth axis acceleration data;a GPS antenna and receiver means mounted in the projectile for providingand updating present position data; a computer and memory mounted in theprojectile and executing a guidance program; and divert propulsionmeans; wherein the guidance program is responsive to the roll, pitch andazimuth axis acceleration data and to the sampled present position datafor calculating and outputting time indexed roll, pitch and azimuthangles, time indexed present position and velocity to a flight controlsystem for guiding the projectile to a predetermined target location;and wherein the divert propulsion means are responsive to the guidanceprogram.
 2. The projectile guidance system of claim 1 wherein saiddivert propulsion means comprises one or more control surfaces locatedexterior to the projectile.
 3. The projectile guidance system of claim 1wherein said divert propulsion means is configured to change theattitude of the projectile in response to a command from the guidanceprogram.
 4. The projectile guidance system of claim 1 wherein saiddivert propulsion means is configured to change the flight path of theprojectile in response to a command from the guidance program.
 5. Theprojectile guidance system of claim 1 wherein said divert propulsionmeans is positioned interior to the projectile.
 6. The projectileguidance system of claim 5 wherein said divert propulsion meanscomprises gas metering means.
 7. The projectile guidance system of claim6 wherein said divert propulsion means comprises gas directing means. 8.The projectile guidance system of claim 7 wherein said gas directingmeans comprises a filter.
 9. The projectile guidance system of claim 6wherein said divert propulsion means comprises a chamber for storingcompressed gas and means for supplying compressed gas to said chamber.10. The projectile guidance system of claim 9 wherein said means forsupplying compressed gas comprising a supply port positioned to receivehigh pressure launch gasses during launch of the projectile, said supplyport configured to permit entry of launch gasses during launch and toretain said launch gasses after launch.
 11. The projectile guidancesystem of claim 10 wherein said gas metering means comprises a filter.12. The projectile guidance system of claim 9 wherein said means forsupplying compressed gas comprises a supply port positioned to receivehigh pressure gasses prior to launch of the projectile, said supply portconfigured to permit entry of high pressure gasses prior to launch andto retain said high pressure gasses after launch.
 13. The projectileguidance system of claim 9 wherein said means for supplying compressedgas comprises a combustion apparatus for combusting solid propellant.14. The projectile guidance system of claim 9 wherein said means forsupplying compressed gas comprises a combustion apparatus for combustingliquid propellant.
 15. The projectile guidance system of claim 9 whereinsaid means for supplying compressed gas comprises a combustion apparatusfor combusting a combination of liquid propellant and solid propellant.16. The projectile guidance system of claim 5 wherein said divertpropulsion means comprises one or more supply ports located on a leadingportion of the projectile.
 17. The projectile guidance system of claim 5wherein said divert propulsion means comprises one or more exit portslocated on a side of the projectile.
 18. The projectile guidance systemof claim 5 wherein said divert propulsion means comprises one or moreexit ports located on a trailing portion of the projectile.
 19. Aprojectile guidance system without gyros comprising: the projectilehaving at least a triax of accelerometers comprising an x accelerometerfor providing x-axis acceleration data measured along the x-axis, anaccelerometer for providing y-axis acceleration data measured along they-axis and a z-accelerometer for providing z-axis acceleration datameasured along the z-axis, a GPS antenna and receiver means forproviding onboard GPS position and GPS velocity data in earth referencednavigational coordinates, a computer and memory and program means forstoring and accessing time indexed GPS position and GPS velocity dataand for transforming the x, y and z axis acceleration data from body tonavigation coordinates, the acceleration data being arrayed and havingtime indexes common with the GPS position and velocity data; and divertpropulsion responsive to the program means; wherein the program means isresponsive to correcting time indexed acceleration data and to GPSvelosity and position data for calculating and outputting an estimatedprojectile roll, pitch and yaw angle with respect to local level foreach time index iteration of present position, velocity and accelerationdata to a flight control system for guiding the projectile to apredetermined target location.
 20. A projectile guidance system processfor the guidance of a vehicle or projectile without gyros, the vehiclehaving been launched and having an orthogonal coordinate system with alongitudinal or x-axis for roll measurement, a y-axis for pitchmeasurement, and a z-axis for yaw measurement, a triax of accelerometersoutputting x, y and z axis acceleration, and a GPS antenna and receivermeans for providing GPS position and GPS velocity data in earthreferenced navigational coordinates, the projectile guidance systemprocess comprising the steps of: receiving, indexing and storing GPSposition and velocity data with concurrent x, y and z axis accelerationdata with a computer and memory means executing a program to calculate atrajectory for the vehicle as a function of a set of state equations,the program converting the x, y and z axis acceleration data to a locallevel navigational reference system, the program then solving a set oftime indexed state equations to update the system's estimated state, andthen computing a time indexed pitch, roll and yaw angle in locally levelcoordinates, the program then outputting the time indexed pitch, rollyaw angle in locally level coordinates with a corresponding time indexedpresent position to a flight control system for guiding the projectileto a destination; the flight control system controlling divertpropulsion means to guide the vehicle or projectile.